...a companion blog to "Math-Frolic," specifically for interviews, book reviews, weekly-linkfests, and longer posts or commentary than usually found at the Math-Frolic site.

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"Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." ---Bertrand Russell (1907) Rob Gluck

"I have come to believe, though very reluctantly, that it [mathematics] consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-legged animal is an animal." ---Bertrand Russell (1957)

******************************************************************** Rob Gluck

Monday, November 24, 2014

Proofiness


Such serendipitous timing (and inspiration) once again from Dr. Keith Devlin. I was in the process of writing a post about "knowing" or "proof" in mathematics... and along comes Dr. D. to finish my post off for me!

First, the preliminary bit I'd been working on:

We'll start with an old joke that most readers are familiar with....

An engineer, a physicist, and a mathematician are riding a train through Scotland, when they look out the window and see a lone black sheep in a field. The engineer remarks, "Hmmm, I guess in Scotland all the sheep are black!" The physicist replies, "No, no, no! Only some Scottish sheep are black." To which the mathematician, rolling his eyes at his fellow travelers' sloppy logic, chimes in, "All one can say is that in Scotland, there is at least one sheep who has at least one side that appears to be black at least some of the time."

As often happens in comedy, the humor stems from an intrinsic kernel of truth... that mathematicians, unlike philosophers, physicists, and others, really are constrained by a stricter regimen of logic and deduction than prevails elsewhere. Of all the sciences, "induction" is probably least acceptable in mathematics, even though reams have been written about the philosophical shortcomings of induction more generally.
One can't conclude just because the first billion values plugged into the Riemann Hypothesis hold true, that therefore the hypothesis IS true, or if the first trillion digits of pi reveal no pattern, that by itself, wouldn't mean there was no pattern to pi. In fact, and sometimes hard for the layperson to comprehend, in mathematics, "billions" and "trillions" and the like, are really very very very very tiny numbers anyway.

A classic example of what mathematicians are up against was the 1885 Mertens Conjecture, which proposed that the sum of the first n values of the Möbius function had an absolute value of, at most, √n. ALL human/computer calculations support the conjecture, but 100 years later, in 1985, Andrew Odlyzko and Herman te Riele theoretically disproved it. So despite the fact that ALL calculations affirm the conjecture, somewhere out there is a "bad" n with a currently known upper bound of e^ (1.59×1040).

[A 2012 "Gödel's Lost Letter" post on  "Apocalypses In Math," including Mertens, is worth reading  http://rjlipton.wordpress.com/2012/12/21/what-would-be-left-if/  ]

You don't have to understand the technical details of the conjecture to sense the giiiiinormousness(!), of that upper bound value. Again, it means that even though the conjecture is TRUE for ALL values ever (so far) plugged into it, it's now known that somewhere out there lies some value for which it is UNtrue.

(And perhaps I should say, lest any reader s'pose that the Mertens Conjecture is some minor, off-the-wall bit of eclectic, unrepresentative math, that IF it had been proven true it would've implied the truth of the Riemann Hypothesis, and all the consequences that flow therefrom.)

Anyway, the word "proof" or "proves" has been an annoying pet peeve of mine across the years. There are NO proofs in science. There is simply the aggregation of evidence... "induction" is certainly rampant, but proof, not so much.  E=mc^2 is NOT proven. Evolution is NOT proven. The existence of the moon, or for that matter my own existence is NOT proven... they just all seem to be the case given our perceptions/interpretations. The philosophical endpoint-conundrum here is that we can't demonstrate conclusively whether-or-not we are anything other than a "brain-in-a-vat," or automaton, completely under the control of a much higher Martian being. What we call "proof" is, at best, a notion residing in the self-enclosed, essentially tautological realm of concocted logic and math.

I once left a comment on a well-known scientist's blog when he wrote about something that was true and proven in physics. I took issue saying that technically it wasn't proven, but simply had a vast preponderance of evidence supporting it (as we perceive the evidence). His response (paraphrasing from memory) was that "Well, of course if you mean 100% absolute metaphysical proof you're right, but nobody seriously uses the term in that sense in everyday parlance, so for all-intents-and-purposes it is proven." And that of course is my beef, that "proof" has been so watered-down, polluted by language and argument, that its precise meaning is lost, and we ought, whenever possible in science, deal in precise, not compromised, meanings.

Recently someone in a Twitter stream stated something to the effect, 'Philosophy deals with ideas, physics with evidence, and only math with proof.'  I came close to re-tweeting it, but in the end didn't feel comfortable enough with it. And one reason it didn't 'feel quite right' has to do with Keith Devlin's latest posting where he takes discomfort with the word "proof" to the next level, essentially saying (and I hope I'm not mis-stating him here) that induction and imprecision inescapably raises its ugly head even when and where we are unaware of it, including mathematics.
Read and savor his entry here:

http://profkeithdevlin.org/2014/11/24/what-is-a-proof-really/

...it has a number of links, and also leads in turn to a secondary, related post about his ongoing MOOC here:

http://mooctalk.org/2014/11/24/how-is-it-going-this-time/

Early on he says, "These days I have a very pragmatic perspective on what a proof is, based on the way people use them in the day-to-day world of mathematics: Proofs are stories that convince suitably qualified others that a certain statement is true."

"Proofs" as "stories" -- I love it, and surely a new way for most folks to wrap their brains around the term. Dr. Devlin spends the rest of the post fleshing out the idea.
 I suspect the average bloke won't get much from Dr. Devlin's message, but for scientists and mathematicians I think it must-reading.

It's been awhile since Dr. Devlin had posted new blog entries. And reading these two pieces from him I almost feel like an addict who was long overdue getting a 'hit' from his drug-of-choice (and I mean that positively! ;-) I feel refreshed and reinvigorated just reading these two posts!

As Keith writes in the beginning: "What is a mathematical proof? Way back when I was a college freshman, I could give you a precise answer... But I was so much older then, I'm younger than that now." ;-) And then he links to the original Dylan version, but I think I'll opt for the smoother voices of the Byrds:




P.S.: so far as I'm aware the word "proofiness" was originally coined by Charles Seife in the title to one of his popular books.

ADDENDUM:  in the course of an email someone mentions that they're not clear what my point is above since it seems like Dr. Devlin's view (of 'proofs as stories') contradicts my initial stance that "proof" is a more stringent term in math than other fields. That's my fault for not making the transition more smoothly: yes, I started this post thinking I would write about why the term "proof"  ought be strictly relegated to mathematics and logic, and NOT used in other sciences nor in everyday parlance... but before I could wrap it up, along came Keith to say that e-e-e-even within mathematics "proof" is an inexact term, not being applied quite as people envision. I thought that was an awesome (and subtle) point and a better wrap-up than what I'd had in mind -- and I don't think it's so much a contradiction to my point, as it is yet a further extension of how loosely "proof" gets bandied about. Hope that helps.


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